Construct Single-Hierarchical P/NBD Model for Short Timeframe Synthetic Data

In this workbook we construct the non-hierarchical P/NBD models on the synthetic data with the longer timeframe.

0.1 Load Short-Timeframe Synthetic Transaction Data

We now want to load the CD-NOW transaction data.

Code

customer_cohortdata_tbl <- read_rds("data/shortsynth_customer_cohort_data_tbl.rds")
customer_cohortdata_tbl |> glimpse()

Rows: 5,000
Columns: 5
$ customer_id     <chr> "SFC202001_0001", "SFC202001_0002", "SFC202001_0003", …
$ cohort_qtr      <chr> "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1",…
$ cohort_ym       <chr> "2020 01", "2020 01", "2020 01", "2020 01", "2020 01",…
$ first_tnx_date  <dttm> 2020-01-01 02:15:58, 2020-01-01 00:40:17, 2020-01-01 …
$ total_tnx_count <int> 3, 9, 1, 2, 5, 1, 4, 6, 2, 4, 1, 11, 3, 3, 1, 1, 11, 3…

Code

customer_transactions_tbl <- read_rds("data/shortsynth_transaction_data_tbl.rds")
customer_transactions_tbl |> glimpse()

Rows: 29,454
Columns: 7
$ customer_id   <fct> SFC202001_0002, SFC202001_0001, SFC202001_0004, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 00:40:17, 2020-01-01 02:15:58, 2020-01-01 18…
$ tnx_dow       <fct> Wed, Wed, Wed, Wed, Thu, Thu, Thu, Thu, Thu, Thu, Thu, F…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0001", "T20200101-0002", "T20200101-0003", "T…
$ tnx_amount    <dbl> 51.73, 139.05, 11.72, 88.30, 1.16, 0.95, 127.85, 35.07, …

Code

customer_subset_id <- read_rds("data/shortsynth_customer_subset_ids.rds")
customer_subset_id |> glimpse()

 Factor w/ 5000 levels "SFC202001_0002",..: 2 3 8 10 14 16 17 21 25 27 ...

We re-produce the visualisation of the transaction times we used in previous workbooks.

Code

plot_tbl <- customer_transactions_tbl |>
  group_nest(customer_id, .key = "cust_data") |>
  filter(map_int(cust_data, nrow) > 3) |>
  slice_sample(n = 30) |>
  unnest(cust_data)

ggplot(plot_tbl, aes(x = tnx_timestamp, y = customer_id)) +
  geom_line() +
  geom_point() +
  labs(
      x = "Date",
      y = "Customer ID",
      title = "Visualisation of Customer Transaction Times"
    ) +
  theme(axis.text.y = element_text(size = 10))

0.2 Load Derived Data

Code

obs_fitdata_tbl   <- read_rds("data/shortsynth_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/shortsynth_obs_validdata_tbl.rds")

customer_fit_stats_tbl <- obs_fitdata_tbl |>
  rename(x = tnx_count)

0.3 Load Subset Data

We also want to construct our data subsets for the purposes of speeding up our valuations.

Code

customer_fit_subset_tbl <- obs_fitdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_fit_subset_tbl |> glimpse()

Rows: 1,000
Columns: 6
$ customer_id    <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC2020…
$ first_tnx_date <dttm> 2020-01-01 02:15:58, 2020-01-01 18:04:32, 2020-01-02 1…
$ last_tnx_date  <dttm> 2020-02-20 14:43:57, 2020-01-10 17:49:12, 2020-07-17 0…
$ tnx_count      <dbl> 2, 1, 5, 4, 0, 10, 10, 0, 0, 0, 6, 2, 0, 0, 0, 1, 0, 3,…
$ t_x            <dbl> 7.2170614, 1.2841927, 28.0650995, 12.3070761, 0.0000000…
$ T_cal          <dbl> 104.4151, 104.3210, 104.1918, 104.1751, 104.0693, 104.0…

Code

customer_valid_subset_tbl <- obs_validdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_valid_subset_tbl |> glimpse()

Rows: 1,000
Columns: 3
$ customer_id       <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC2…
$ tnx_count         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ tnx_last_interval <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, …

We now use these datasets to set the start and end dates for our various validation methods.

Code

dates_lst <- read_rds("data/shortsynth_simulation_dates.rds")

use_fit_start_date <- dates_lst$use_fit_start_date
use_fit_end_date   <- dates_lst$use_fit_end_date

use_valid_start_date <- dates_lst$use_valid_start_date
use_valid_end_date   <- dates_lst$use_valid_end_date

We now split out the transaction data into fit and validation datasets.

Code

customer_fit_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_fit_start_date,
    tnx_timestamp <= use_fit_end_date
    )
  
customer_fit_transactions_tbl |> glimpse()

Rows: 5,016
Columns: 7
$ customer_id   <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 02:15:58, 2020-01-01 18:04:32, 2020-01-02 15…
$ tnx_dow       <fct> Wed, Wed, Thu, Thu, Fri, Fri, Fri, Sat, Sat, Sun, Sun, M…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0002", "T20200101-0003", "T20200102-0004", "T…
$ tnx_amount    <dbl> 139.05, 11.72, 35.07, 54.66, 70.30, 91.17, 393.76, 588.8…

Code

customer_valid_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_valid_start_date,
    tnx_timestamp <= use_valid_end_date
    )
  
customer_valid_transactions_tbl |> glimpse()

Rows: 1,509
Columns: 7
$ customer_id   <fct> SFC202108_0025, SFC202008_0057, SFC202112_0120, SFC20211…
$ tnx_timestamp <dttm> 2022-01-01 00:01:40, 2022-01-01 01:06:39, 2022-01-01 05…
$ tnx_dow       <fct> Sat, Sat, Sat, Sat, Sat, Sat, Sat, Sat, Sun, Sun, Sun, S…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20220101-0001", "T20220101-0002", "T20220101-0007", "T…
$ tnx_amount    <dbl> 2.97, 69.73, 0.36, 4.88, 725.98, 22.79, 45.67, 0.60, 8.1…

Finally, we want to extract the first transaction for each customer, so we can add this data to assess our models.

Code

customer_initial_tnx_tbl <- customer_fit_transactions_tbl |>
  slice_min(n = 1, order_by = tnx_timestamp, by = customer_id)

customer_initial_tnx_tbl |> glimpse()

Rows: 1,000
Columns: 7
$ customer_id   <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 02:15:58, 2020-01-01 18:04:32, 2020-01-02 15…
$ tnx_dow       <fct> Wed, Wed, Thu, Thu, Fri, Fri, Fri, Sat, Sat, Sun, Sun, M…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0002", "T20200101-0003", "T20200102-0004", "T…
$ tnx_amount    <dbl> 139.05, 11.72, 35.07, 54.66, 70.30, 91.17, 393.76, 588.8…

We now expand out these initial transactions so that we can append them to our simulations.

Code

sim_init_tbl <- customer_initial_tnx_tbl |>
  transmute(
    customer_id,
    draw_id       = list(1:n_sim),
    tnx_timestamp,
    tnx_amount
    ) |>
  unnest(draw_id)

sim_init_tbl |> glimpse()

Rows: 2,000,000
Columns: 4
$ customer_id   <fct> SFC202001_0001, SFC202001_0001, SFC202001_0001, SFC20200…
$ draw_id       <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
$ tnx_timestamp <dttm> 2020-01-01 02:15:58, 2020-01-01 02:15:58, 2020-01-01 02…
$ tnx_amount    <dbl> 139.05, 139.05, 139.05, 139.05, 139.05, 139.05, 139.05, …

Before we start on that, we set a few parameters for the workbook to organise our Stan code.

Code

stan_modeldir <- "stan_models"
stan_codedir  <-   "stan_code"

1 Fit First Hierarchical Lambda Model

Our first hierarchical model puts a hierarchical prior around the mean of our population \(\lambda\) - lambda_mn.

Once again we use a Gamma prior for it.

1.1 Compile and Fit Stan Model

We now compile this model using CmdStanR.

Code

pnbd_onehierlambda_stanmodel <- cmdstan_model(
  "stan_code/pnbd_onehier_lambda.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_short_onehierlambda1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_lambda_mn_p1 = 0.25,
    hier_lambda_mn_p2 = 1,

    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_onehierlambda1_stanfit <- pnbd_onehierlambda_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_short_onehierlambda1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_onehierlambda1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_onehierlambda1_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -46902.31 -46899.75 64.07 62.94 -47012.20 -46796.38 1.00      546
 lambda_mn      0.25      0.25  0.01  0.01      0.24      0.26 1.00     1518
 lambda[1]      0.44      0.42  0.15  0.14      0.22      0.70 1.00     2862
 lambda[2]      0.21      0.18  0.13  0.12      0.05      0.47 1.00     2869
 lambda[3]      0.26      0.21  0.21  0.16      0.04      0.64 1.00     2586
 lambda[4]      0.14      0.08  0.16  0.09      0.00      0.44 1.00     1750
 lambda[5]      0.49      0.44  0.27  0.24      0.15      0.99 1.00     2470
 lambda[6]      0.24      0.19  0.20  0.15      0.03      0.62 1.00     1973
 lambda[7]      0.14      0.08  0.17  0.09      0.01      0.49 1.00     2590
 lambda[8]      0.16      0.15  0.07  0.07      0.07      0.30 1.00     2094
 ess_tail
      812
     1600
     1118
     1324
     1163
      714
     1479
     1356
     1113
     1049

 # showing 10 of 9963 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_onehierlambda1_stanfit$cmdstan_diagnose()

File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-1.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-2.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-3.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda1-4.csv not found
No valid input files, exiting.

1.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_onehierlambda1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_short_onehierlambda1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_onehierlambda1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

1.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_short_onehierlambda1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_onehierlambda1_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_short_onehierlambda1",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 5110
  )

pnbd_short_onehierlambda1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_onehierlambda1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_onehierlambda1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_onehierlambda1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_onehierlambda1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_onehierlambda1_assess_valid_simstats_tbl.rds"

1.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_onehierlambda1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

1.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_onehierlambda1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

2 Fit Second Hierarchical Lambda Model

In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.

2.1 Fit Stan Model

We now want to fit the model to our data using this alternative prior for lambda_mn.

Code

stan_modelname <- "pnbd_short_onehierlambda2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_lambda_mn_p1 = 0.50,
    hier_lambda_mn_p2 = 1,

    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_onehierlambda2_stanfit <- pnbd_onehierlambda_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_short_onehierlambda2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_onehierlambda2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_onehierlambda2_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -46898.74 -46898.85 62.46 63.38 -47002.80 -46798.39 1.01      621
 lambda_mn      0.25      0.25  0.01  0.01      0.24      0.26 1.00     1375
 lambda[1]      0.44      0.42  0.15  0.15      0.23      0.71 1.00     2541
 lambda[2]      0.21      0.18  0.14  0.12      0.05      0.47 1.00     2095
 lambda[3]      0.26      0.21  0.20  0.17      0.04      0.67 1.00     1712
 lambda[4]      0.14      0.08  0.18  0.09      0.00      0.50 1.00     1890
 lambda[5]      0.49      0.44  0.27  0.25      0.14      1.01 1.00     2187
 lambda[6]      0.25      0.19  0.20  0.15      0.03      0.64 1.00     2136
 lambda[7]      0.14      0.08  0.17  0.09      0.01      0.45 1.00     1787
 lambda[8]      0.16      0.15  0.07  0.07      0.07      0.29 1.00     2572
 ess_tail
     1059
     1347
     1522
     1273
     1138
     1041
     1277
      815
     1047
     1237

 # showing 10 of 9963 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_onehierlambda2_stanfit$cmdstan_diagnose()

File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-1.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-2.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-3.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehierlambda2-4.csv not found
No valid input files, exiting.

2.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_onehierlambda2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_short_onehierlambda2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_onehierlambda2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

2.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_short_onehierlambda2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_onehierlambda2_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_short_onehierlambda2",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 5120
  )

pnbd_short_onehierlambda2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_onehierlambda2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_onehierlambda2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_onehierlambda2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_onehierlambda2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_onehierlambda2_assess_valid_simstats_tbl.rds"

2.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_onehierlambda2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

2.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_onehierlambda2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

3 Fit First Hierarchical Mu Model

We now construct the same hierarchical model but based around mu_mn.

3.1 Compile and Fit Stan Model

We compile this model using CmdStanR.

Code

pnbd_onehiermu_stanmodel <- cmdstan_model(
  "stan_code/pnbd_onehier_mu.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_short_onehiermu1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_mu_mn_p1 = 0.50,
    hier_mu_mn_p2 = 1.00,

    lambda_mn     = 0.25,
    lambda_cv     = 1.00,

    mu_cv         = 1.00

    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_onehiermu1_stanfit <- pnbd_onehiermu_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_short_onehiermu1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_onehiermu1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_onehiermu1_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -43842.72 -43843.00 61.97 64.34 -43944.91 -43744.00 1.01      682
 mu_mn          0.11      0.11  0.00  0.00      0.11      0.12 1.01      584
 lambda[1]      0.44      0.43  0.15  0.15      0.23      0.73 1.00     2774
 lambda[2]      0.21      0.18  0.13  0.12      0.05      0.47 1.00     2111
 lambda[3]      0.27      0.22  0.20  0.17      0.04      0.66 1.00     2428
 lambda[4]      0.14      0.09  0.15  0.10      0.00      0.44 1.00     1672
 lambda[5]      0.49      0.45  0.26  0.24      0.16      0.99 1.00     1977
 lambda[6]      0.25      0.20  0.20  0.15      0.04      0.61 1.00     2000
 lambda[7]      0.14      0.08  0.16  0.10      0.00      0.46 1.00     1493
 lambda[8]      0.16      0.15  0.07  0.06      0.07      0.29 1.00     2761
 ess_tail
     1123
      864
     1462
     1342
      967
      839
     1005
     1266
      794
     1468

 # showing 10 of 9963 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_onehiermu1_stanfit$cmdstan_diagnose()

File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-1.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-2.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-3.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu1-4.csv not found
No valid input files, exiting.

3.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "mu_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_onehiermu1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_short_onehiermu1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_onehiermu1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

3.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_short_onehiermu1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_onehiermu1_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_short_onehiermu1",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 5130
  )

pnbd_short_onehiermu1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_onehiermu1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_onehiermu1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_onehiermu1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_onehiermu1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_onehiermu1_assess_valid_simstats_tbl.rds"

3.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_onehiermu1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

3.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_onehiermu1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

4 Fit Second Hierarchical Mu Model

In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.

4.1 Fit Stan Model

We now want to fit the model to our data using this alternative prior for lambda_mn.

Code

stan_modelname <- "pnbd_short_onehiermu2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed   <- stanfit_seed + 1

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    hier_mu_mn_p1 = 0.25,
    hier_mu_mn_p2 = 1.00,

    lambda_mn     = 0.25,
    lambda_cv     = 1.00,

    mu_cv         = 1.00
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_onehiermu2_stanfit <- pnbd_onehiermu_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_short_onehiermu2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_onehiermu2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_onehiermu2_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -43841.15 -43840.60 64.89 66.87 -43948.90 -43736.50 1.01      593
 mu_mn          0.11      0.11  0.00  0.00      0.11      0.12 1.01      384
 lambda[1]      0.45      0.43  0.15  0.15      0.23      0.74 1.00     3278
 lambda[2]      0.21      0.18  0.13  0.12      0.06      0.46 1.01     2522
 lambda[3]      0.26      0.21  0.21  0.17      0.04      0.68 1.00     1921
 lambda[4]      0.15      0.08  0.17  0.10      0.01      0.49 1.00     1712
 lambda[5]      0.49      0.44  0.27  0.25      0.15      1.00 1.00     2035
 lambda[6]      0.24      0.20  0.20  0.15      0.03      0.62 1.01     2510
 lambda[7]      0.14      0.08  0.15  0.09      0.01      0.45 1.00     1927
 lambda[8]      0.16      0.16  0.07  0.06      0.07      0.29 1.00     3151
 ess_tail
      887
      840
     1273
     1367
     1140
     1023
     1183
     1104
     1091
     1176

 # showing 10 of 9963 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_onehiermu2_stanfit$cmdstan_diagnose()

File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-1.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-2.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-3.csv not found
File /home/rstudio/btydwork/stan_models/fit_pnbd_init_short_onehiermu2-4.csv not found
No valid input files, exiting.

4.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "mu_mn",
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_onehiermu2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_short_onehiermu2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_onehiermu2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

4.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

Code

pnbd_stanfit <- pnbd_short_onehiermu2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_onehiermu2_assess_data_lst <- run_model_assessment(
  model_stanfit    = pnbd_stanfit,
  insample_tbl     = customer_fit_subset_tbl,
  fit_label        = "pnbd_short_onehiermu2",
  fit_end_dttm     = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm   = use_valid_end_date   |> as.POSIXct(),
  sim_seed         = 5140
  )

pnbd_short_onehiermu2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_onehiermu2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_onehiermu2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_onehiermu2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_onehiermu2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_onehiermu2_assess_valid_simstats_tbl.rds"

4.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_onehiermu2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

4.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_onehiermu2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

As for our short time frame data, overall our model is working well.

5 Compare Model Outputs

We have looked at each of the models individually, but it is also worth looking at each of the models as a group.

We now want to combine both the fit and valid transaction sets to calculate the summary statistics for both.

Code

obs_summstats_tbl <- list(
    fit   = customer_fit_transactions_tbl,
    valid = customer_valid_transactions_tbl
    ) |>
  bind_rows(.id = "assess_type") |>
  group_by(assess_type) |>
  calculate_transaction_summary_statistics() |>
  pivot_longer(
    cols      = !assess_type,
    names_to  = "label",
    values_to = "obs_value"
    )

obs_summstats_tbl |> glimpse()

Rows: 16
Columns: 3
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "v…
$ label       <chr> "p10", "p25", "p50", "p75", "p90", "p99", "total_count", "…
$ obs_value   <dbl> 1.00000, 1.00000, 2.00000, 5.00000, 12.00000, 45.01000, 50…

Code

model_assess_transactions_tbl <- dir_ls("data", regexp = "pnbd_short_(fixed|one).*_assess_.*index") |>
  enframe(name = NULL, value = "file_path") |>
  mutate(
    model_label = str_replace(file_path, "data/pnbd_short_(.*?)_assess_.*", "\\1"),
    assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
    
    assess_data = map(
      file_path, construct_model_assessment_data,
      
      .progress = "construct_assess_data"
      )
    ) |>
  select(model_label, assess_type, assess_data) |>
  unnest(assess_data)

model_assess_transactions_tbl |> glimpse()

Rows: 75,039,662
Columns: 6
$ model_label   <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed…
$ assess_type   <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", …
$ customer_id   <fct> SFC202001_0001, SFC202001_0001, SFC202001_0001, SFC20200…
$ draw_id       <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 4, 4, 4, 4, 4, 4, 4,…
$ tnx_timestamp <dttm> 2020-02-27 04:24:25, 2020-03-23 07:41:18, 2020-07-02 04…
$ tnx_amount    <dbl> 83.09, 0.73, 79.75, 33.17, 23.00, 89.13, 198.72, 22.91, …

We now want to calculate the transaction statistics on this full dataset, for each separate draw.

Code

model_assess_tbl <- model_assess_transactions_tbl |>
  group_by(model_label, assess_type, draw_id) |>
  calculate_transaction_summary_statistics()

model_assess_tbl |> glimpse()

Rows: 32,000
Columns: 11
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed1"…
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "f…
$ draw_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,…
$ p10         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p25         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p50         <dbl> 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3…
$ p75         <dbl> 8, 8, 7, 9, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 8, 8…
$ p90         <dbl> 14.4, 18.0, 16.6, 19.0, 15.0, 16.0, 17.0, 17.0, 18.0, 18.0…
$ p99         <dbl> 57.00, 57.40, 46.32, 53.00, 56.82, 49.06, 44.84, 48.00, 49…
$ total_count <int> 4195, 4242, 4358, 4474, 4320, 4006, 4090, 4146, 4161, 4441…
$ mean_count  <dbl> 6.911038, 7.177665, 6.862992, 7.444260, 7.081967, 6.698997…

We now combine all this data to create a number of different comparison plots for the various summary statistics.

Code

#! echo: TRUE

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "total_count", "Total Transactions"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "mean_count", "Average Transactions per Customer"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "p99", "99th Percentile Count"
  )

5.1 Write Assessment Data to Disk

We now want to save the assessment data to disk.

Code

model_assess_tbl |> write_rds("data/assess_data_pnbd_short_onehier_tbl.rds")

R Environment

Code

options(width = 120L)
sessioninfo::session_info()

─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
 setting  value
 version  R version 4.3.1 (2023-06-16)
 os       Ubuntu 22.04.3 LTS
 system   x86_64, linux-gnu
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       Europe/Dublin
 date     2023-11-20
 pandoc   3.1.1 @ /usr/local/bin/ (via rmarkdown)

─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
 package        * version    date (UTC) lib source
 abind            1.4-5      2016-07-21 [1] RSPM (R 4.3.0)
 arrayhelpers     1.1-0      2020-02-04 [1] RSPM (R 4.3.0)
 backports        1.4.1      2021-12-13 [1] RSPM (R 4.3.0)
 base64enc        0.1-3      2015-07-28 [1] RSPM (R 4.3.0)
 bayesplot      * 1.10.0     2022-11-16 [1] RSPM (R 4.3.0)
 bit              4.0.5      2022-11-15 [1] RSPM (R 4.3.0)
 bit64            4.0.5      2020-08-30 [1] RSPM (R 4.3.0)
 bridgesampling   1.1-2      2021-04-16 [1] RSPM (R 4.3.0)
 brms           * 2.20.4     2023-09-25 [1] RSPM (R 4.3.0)
 Brobdingnag      1.2-9      2022-10-19 [1] RSPM (R 4.3.0)
 cachem           1.0.8      2023-05-01 [1] RSPM (R 4.3.0)
 callr            3.7.3      2022-11-02 [1] RSPM (R 4.3.0)
 checkmate        2.3.0      2023-10-25 [1] RSPM (R 4.3.0)
 cli              3.6.1      2023-03-23 [1] RSPM (R 4.3.0)
 cmdstanr       * 0.6.0.9000 2023-11-07 [1] Github (stan-dev/cmdstanr@a13c798)
 coda             0.19-4     2020-09-30 [1] RSPM (R 4.3.0)
 codetools        0.2-19     2023-02-01 [2] CRAN (R 4.3.1)
 colorspace       2.1-0      2023-01-23 [1] RSPM (R 4.3.0)
 colourpicker     1.3.0      2023-08-21 [1] RSPM (R 4.3.0)
 conflicted     * 1.2.0      2023-02-01 [1] RSPM (R 4.3.0)
 cowplot        * 1.1.1      2020-12-30 [1] RSPM (R 4.3.0)
 crayon           1.5.2      2022-09-29 [1] RSPM (R 4.3.0)
 crosstalk        1.2.0      2021-11-04 [1] RSPM (R 4.3.0)
 curl             5.1.0      2023-10-02 [1] RSPM (R 4.3.0)
 digest           0.6.33     2023-07-07 [1] RSPM (R 4.3.0)
 directlabels   * 2023.8.25  2023-09-01 [1] RSPM (R 4.3.0)
 distributional   0.3.2      2023-03-22 [1] RSPM (R 4.3.0)
 dplyr          * 1.1.3      2023-09-03 [1] RSPM (R 4.3.0)
 DT               0.30       2023-10-05 [1] RSPM (R 4.3.0)
 dygraphs         1.1.1.6    2018-07-11 [1] RSPM (R 4.3.0)
 ellipsis         0.3.2      2021-04-29 [1] RSPM (R 4.3.0)
 evaluate         0.22       2023-09-29 [1] RSPM (R 4.3.0)
 fansi            1.0.5      2023-10-08 [1] RSPM (R 4.3.0)
 farver           2.1.1      2022-07-06 [1] RSPM (R 4.3.0)
 fastmap          1.1.1      2023-02-24 [1] RSPM (R 4.3.0)
 forcats        * 1.0.0      2023-01-29 [1] RSPM (R 4.3.0)
 fs             * 1.6.3      2023-07-20 [1] RSPM (R 4.3.0)
 furrr          * 0.3.1      2022-08-15 [1] RSPM (R 4.3.0)
 future         * 1.33.0     2023-07-01 [1] RSPM (R 4.3.0)
 generics         0.1.3      2022-07-05 [1] RSPM (R 4.3.0)
 ggdist           3.3.0      2023-05-13 [1] RSPM (R 4.3.0)
 ggplot2        * 3.4.4      2023-10-12 [1] RSPM (R 4.3.0)
 globals          0.16.2     2022-11-21 [1] RSPM (R 4.3.0)
 glue           * 1.6.2      2022-02-24 [1] RSPM (R 4.3.0)
 gridExtra        2.3        2017-09-09 [1] RSPM (R 4.3.0)
 gtable           0.3.4      2023-08-21 [1] RSPM (R 4.3.0)
 gtools           3.9.4      2022-11-27 [1] RSPM (R 4.3.0)
 hms              1.1.3      2023-03-21 [1] RSPM (R 4.3.0)
 htmltools        0.5.6.1    2023-10-06 [1] RSPM (R 4.3.0)
 htmlwidgets      1.6.2      2023-03-17 [1] RSPM (R 4.3.0)
 httpuv           1.6.12     2023-10-23 [1] RSPM (R 4.3.0)
 igraph           1.5.1      2023-08-10 [1] RSPM (R 4.3.0)
 inline           0.3.19     2021-05-31 [1] RSPM (R 4.3.0)
 jsonlite         1.8.7      2023-06-29 [1] RSPM (R 4.3.0)
 knitr            1.44       2023-09-11 [1] RSPM (R 4.3.0)
 labeling         0.4.3      2023-08-29 [1] RSPM (R 4.3.0)
 later            1.3.1      2023-05-02 [1] RSPM (R 4.3.0)
 lattice          0.21-8     2023-04-05 [2] CRAN (R 4.3.1)
 lifecycle        1.0.3      2022-10-07 [1] RSPM (R 4.3.0)
 listenv          0.9.0      2022-12-16 [1] RSPM (R 4.3.0)
 loo              2.6.0      2023-03-31 [1] RSPM (R 4.3.0)
 lubridate      * 1.9.3      2023-09-27 [1] RSPM (R 4.3.0)
 magrittr       * 2.0.3      2022-03-30 [1] RSPM (R 4.3.0)
 markdown         1.11       2023-10-19 [1] RSPM (R 4.3.0)
 Matrix           1.5-4.1    2023-05-18 [2] CRAN (R 4.3.1)
 matrixStats      1.0.0      2023-06-02 [1] RSPM (R 4.3.0)
 memoise          2.0.1      2021-11-26 [1] RSPM (R 4.3.0)
 mime             0.12       2021-09-28 [1] RSPM (R 4.3.0)
 miniUI           0.1.1.1    2018-05-18 [1] RSPM (R 4.3.0)
 munsell          0.5.0      2018-06-12 [1] RSPM (R 4.3.0)
 mvtnorm          1.2-3      2023-08-25 [1] RSPM (R 4.3.0)
 nlme             3.1-162    2023-01-31 [2] CRAN (R 4.3.1)
 parallelly       1.36.0     2023-05-26 [1] RSPM (R 4.3.0)
 pillar           1.9.0      2023-03-22 [1] RSPM (R 4.3.0)
 pkgbuild         1.4.2      2023-06-26 [1] RSPM (R 4.3.0)
 pkgconfig        2.0.3      2019-09-22 [1] RSPM (R 4.3.0)
 plyr             1.8.9      2023-10-02 [1] RSPM (R 4.3.0)
 posterior      * 1.4.1      2023-03-14 [1] RSPM (R 4.3.0)
 prettyunits      1.2.0      2023-09-24 [1] RSPM (R 4.3.0)
 processx         3.8.2      2023-06-30 [1] RSPM (R 4.3.0)
 promises         1.2.1      2023-08-10 [1] RSPM (R 4.3.0)
 ps               1.7.5      2023-04-18 [1] RSPM (R 4.3.0)
 purrr          * 1.0.2      2023-08-10 [1] RSPM (R 4.3.0)
 quadprog         1.5-8      2019-11-20 [1] RSPM (R 4.3.0)
 QuickJSR         1.0.7      2023-10-15 [1] RSPM (R 4.3.0)
 R6               2.5.1      2021-08-19 [1] RSPM (R 4.3.0)
 Rcpp           * 1.0.11     2023-07-06 [1] RSPM (R 4.3.0)
 RcppParallel     5.1.7      2023-02-27 [1] RSPM (R 4.3.0)
 readr          * 2.1.4      2023-02-10 [1] RSPM (R 4.3.0)
 reshape2         1.4.4      2020-04-09 [1] RSPM (R 4.3.0)
 rlang          * 1.1.1      2023-04-28 [1] RSPM (R 4.3.0)
 rmarkdown        2.25       2023-09-18 [1] RSPM (R 4.3.0)
 rstan            2.32.3     2023-10-15 [1] RSPM (R 4.3.0)
 rstantools       2.3.1.1    2023-07-18 [1] RSPM (R 4.3.0)
 rstudioapi       0.15.0     2023-07-07 [1] RSPM (R 4.3.0)
 rsyslog        * 1.0.3      2023-05-08 [1] RSPM (R 4.3.0)
 scales         * 1.2.1      2022-08-20 [1] RSPM (R 4.3.0)
 sessioninfo      1.2.2      2021-12-06 [1] RSPM (R 4.3.0)
 shiny            1.7.5.1    2023-10-14 [1] RSPM (R 4.3.0)
 shinyjs          2.1.0      2021-12-23 [1] RSPM (R 4.3.0)
 shinystan        2.6.0      2022-03-03 [1] RSPM (R 4.3.0)
 shinythemes      1.2.0      2021-01-25 [1] RSPM (R 4.3.0)
 StanHeaders      2.26.28    2023-09-07 [1] RSPM (R 4.3.0)
 stringi          1.7.12     2023-01-11 [1] RSPM (R 4.3.0)
 stringr        * 1.5.0      2022-12-02 [1] RSPM (R 4.3.0)
 svUnit           1.0.6      2021-04-19 [1] RSPM (R 4.3.0)
 tensorA          0.36.2     2020-11-19 [1] RSPM (R 4.3.0)
 threejs          0.3.3      2020-01-21 [1] RSPM (R 4.3.0)
 tibble         * 3.2.1      2023-03-20 [1] RSPM (R 4.3.0)
 tidybayes      * 3.0.6      2023-08-12 [1] RSPM (R 4.3.0)
 tidyr          * 1.3.0      2023-01-24 [1] RSPM (R 4.3.0)
 tidyselect       1.2.0      2022-10-10 [1] RSPM (R 4.3.0)
 tidyverse      * 2.0.0      2023-02-22 [1] RSPM (R 4.3.0)
 timechange       0.2.0      2023-01-11 [1] RSPM (R 4.3.0)
 tzdb             0.4.0      2023-05-12 [1] RSPM (R 4.3.0)
 utf8             1.2.4      2023-10-22 [1] RSPM (R 4.3.0)
 V8               4.4.0      2023-10-09 [1] RSPM (R 4.3.0)
 vctrs            0.6.4      2023-10-12 [1] RSPM (R 4.3.0)
 vroom            1.6.4      2023-10-02 [1] RSPM (R 4.3.0)
 withr            2.5.1      2023-09-26 [1] RSPM (R 4.3.0)
 xfun             0.40       2023-08-09 [1] RSPM (R 4.3.0)
 xtable           1.8-4      2019-04-21 [1] RSPM (R 4.3.0)
 xts              0.13.1     2023-04-16 [1] RSPM (R 4.3.0)
 yaml             2.3.7      2023-01-23 [1] RSPM (R 4.3.0)
 zoo              1.8-12     2023-04-13 [1] RSPM (R 4.3.0)

 [1] /usr/local/lib/R/site-library
 [2] /usr/local/lib/R/library

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Code

options(width = 80L)